A Trust Model in Bootstrap Percolation

Rinni Bhansali and Laura P. Schaposnik

1

Abstract

Bootstrap percolation is a class of monotone cellular automata describing an activation process whichfollows certain activation rules. In particular, in the classical r-neighbor bootstrap process on a graphG, a set A of initially infected vertices spreads by infecting vertices with at least r already-infectedneighbors. Motivated by the study of social networks and biological interactions through graphs, wherevertices represent people and edges represent the relations amongst them, we introduce here a novelmodel which we name T -bootstrap percolation (T -BP). In this new model, vertices of the graph G areassigned random labels, and the set of initially infected vertices spreads by infecting (at each time step)vertices with at least a fixed number of already-infected neighbors of each label.

The Trust Model for Bootstrap Percolation allows one to impose a preset level of skepticism towardsa rumor, as it requires a rumor to be validated by numerous groups in order for it to spread, henceimposing a predetermined level of trust needed for the rumor to spread. By considering different randomand non-random networks, we describe various properties of this new model (e.g., the critical probabilityof infection and the confidence threshold), and compare it to other types of bootstrap percolation fromthe literature, such as U -bootstrap percolation. Ultimately, we describe its implications when applied torumor spread, fake news, and marketing strategies, along with potential future applications in modelingthe spread of genetic diseases.

2

Contents

1 Introduction 41.1 Defining the T -BP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Summary of main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Bootstrap Percolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3.1 Classical Bootstrap Percolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3.2 U -bootstrap percolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Other Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 A Trust Model for Bootstrap Percolation 72.1 Immunity in T -BP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 Immunity for T -BP as r-bootstrap percolation . . . . . . . . . . . . . . . . . . . . 82.1.2 Immunity for the T -BP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.1.3 A Network’s Immunity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Immunity on Hierarchical Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3 Critical Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Two-Dimensional Trust Model for Bootstrap Percolation 133.1 Critical probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 T -BP as U -Bootstrap Percolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2.1 Striped T -BP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2.2 Diagonal BP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.3 Other two dimensional T -BP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.3.1 3-1 BP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.3.2 Checkerboard BP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3.3 Mixed BP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.4 Trust Model for T -BP for m = 2 as 2-bootstrap percolation on Z2 . . . . . . . . . . . . . . 18

4 T -BP on Random Networks 194.1 Varying Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.2 Varying the number of required labels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5 Conclusion 20

6 Acknowledgements 21

7 References 22

3

1 IntroductionSooner or later most people struggle to find the perfect present: we hear from our son’s friends that certainremote-controlled helicopter would be so fun for his 10th birthday – but will it be safe? Once we hear fromour son’s friends’ parents that the helicopter is indeed entertaining and safe for his age, we are closer tobuying it. Is this recurrent phenomenon a consequence of a natural instinct that one has, where having thesame information transmitted by different “types” of people makes it become more trustworthy? If so, wemay wonder naturally:

How many different types of people (colleagues, friends, Uber drivers, etc.) should we hear a piece of newsfrom, before we start transmitting it as a true fact?

Figure 1: Examples of asocial network where dif-ferent colors–labels–describethe different types of peopleneeded for gossip spreading.

Requiring a clear range for sources of information would allow membersof a society to express skepticism towards gossip and fake news. Moreover,understanding this range could allow various industries to target wisely a min-imum amount of consumers within each type of people, and use the naturalspreading process of information to continue the marketing on its own.

Bootstrap percolation is a particular class of monotone cellular automatadescribing an activation process which follows certain activation rules, andwhich has been used to model interactions within societies. In particular, inthe classical r-neighbor bootstrap process on a graph G, a set A of initially“infected” vertices spreads by infecting vertices with at least r already-infectedneighbors. We shall introduce here a modified version of this model, which wecall Trust Model in Bootstrap Percolation (T -BP), designed to answer an equivalent question to those posedabove:

How does information percolate when a messenger only passes the information on if it has been receivedfrom a number of different sources of certain types?

1.1 Defining the T -BPConsider a finite or infinite graph G, a number m ∈ N, a trust vector K = (k1, . . . ,km) ∈ Nm of non-negativenumbers, and a set A := A0 of initially “infected” vertices in G. After assigning randomly a label in{1, . . . ,m} to each vertex, we define bootstrap percolation with trust vector K (this is, a trust model forbootstrap percolation with level K) as the process where at each time, all of the vertices which have at leastki adjacent vertices infected with label i become infected. Hence, at each time t, the set of infected verticesis given by

At+1 = At ∪{v ∈V (G) : |Ni(v)∩At | ≥ ki for i = 1, . . . ,m},

where Ni(v) denotes the set of adjacent vertices to v in G with label i.Yet while it is of interest to study bootstrap percolation for a fixed trust vector K, it is also important to

consider models where families of trust vectors are allowed at the same time. Following the idea of an updatefamily U from [8], as described in Section 1.3.2, we shall consider a trust family T := {K1, . . . ,Kn} of trust

4

vectors K j = (k j1, . . . ,k

jm) ∈ Nm. Then, we shall refer to Trusted Bootstrap Percolation, or T -bootstrap

percolation (T -BP), as the percolation process where A0 := A and

At+1 = At ∪{v ∈V (G) : ∃K j ∈T s.t. |Ni(v)∩At | ≥ k ji for i = 1, . . . ,m}. (1)

At+1 = At ∪{v ∈V (G) : ∃K j ∈T s.t. |Ni(v)∩At | ≥ k ji for i = 1, . . . ,m}. (2)

Additionally, denote by f : V (G)→ 1, ...,m the function that assigns the random labels to each vertex of

G and note that N(V ) =m⋃

i=1Ni(V ).

1.2 Summary of main resultsAlong the paper we study the above new model, T -bootstrap percolation, from different perspectives. First,in Section 2, we look at immunity of vertices, where a vertex v is immune if for v 6∈ A0, then v 6∈ A∞. Wecompute the probability that a vertex v of degree d is immune for the following models:• T -BP as r-bootstrap percolation (see Proposition 7). Consider T -BP on a graph G with multiple

trust vectors of length m whose entries are k j ∈ {0,1}, with exactly r non-zero.• T -BP with single trust vector K (see Proposition 14). Consider T -BP on a graph G with a single

trust vector K with entries (k1,k2, . . . ,km).Inspired by the regular structure of natural systems in chemistry and biology, in Section 3 we study

the T -bootstrap percolation on the following two dimensional regular networks which can or can not beunderstood through U -bootstrap percolation:• Striped lattice (A case of U -bootstrap percolation, see Section 3.2.1)• Diagonal lattice (A case of U -bootstrap percolation, see Section 3.2.2)• Three-to-One lattice (A case of non-U -bootstrap percolation, see Section 3.3.1)• Checkerboard lattice (A case of non-U -bootstrap percolation, see Section 3.3.2)• Mixed lattice (A case of non-U -bootstrap percolation, see Section 3.3.3)Finally, since social networks are best modeled by random graphs with certain particular density prop-

erties, we dedicate Section 4 to the numerical study of the T -BP on random networks.

1.3 Bootstrap PercolationBootstrap percolation was first introduced in the context of disordered magnetic systems in 1979 by Chalupa,Leath and Reich [10], where the classical r-neighbor bootstrap process on a graph G was studied. Since then,this process has been investigated on several different kinds of networks including finite grids [1, 3, 12, 13],trees [4], and random networks [5, 14], and with many different applications in fields such as sociology[11, 20], physics [17], and epidemiology [7].

1.3.1 Classical Bootstrap PercolationIt is important to note that classical r-neighbor bootstrap percolation is a particular case of T -bootstrappercolation. In particular, one may consider m = 1, and set T = {K1 = (r)} to recover r-neighbor bootstrappercolation with only one label, in which if an uninfected vertex has r infected neighbors, it will becomeinfected in the next time step.

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1.3.2 U -bootstrap percolationRecently Bollobas, Smith and Uzzell introduced in [8] a class of percolation models called U -bootstrappercolation, which includes r-neighbor bootstrap percolation as a special case. These models have certainupdate rules encoded in a set U , and in the case of two dimensional lattices, they divided the model intothree classes: subcritical, critical and supercritical. In what follows, we shall describe these models and inthe following sections, we shall study different types of T -BP and describe when they can be obtained asparticular cases of U -bootstrap percolation.

As in the various bootstrap percolation models, consider a graph G with vertex set V (G). For U -bootstrap percolation in particular, let G = Z2. Then, consider a set A0 ⊂V (G) of initially infected vertices,where each vertex v ∈ V (G) has probability p of being in A0. In order to determine how this initial in-fection spreads, consider a finite collection of finite, non-empty subsets U = {X1,X2, ...,Xn} of G. Giventhe above, in a U -bootstrap percolation process on G, the set At of infected vertices at time t is defined asAt+1 = At

⋃{x ∈ G | there exists i ∈ [n] such that Xi + x ∈ At}, where A0 := A is the set of initially infected

vertices, the set U is referred to as the update family, and each Xi as an update rule.

Remark 1. One can see that r-neighbor bootstrap percolation on Zd is an example of a U -bootstrap perco-lation model, where the update rules are the

(2dr

)r-subsets of the neighbors of the origin.

Along the paper we shall write [A] for the set of all vertices that eventually become infected [A] =∞⋃

t=0At ,

and say that A is closed under U if we have [A] =A. Following the notation of [8], we consider the unit circleS1 ∈ R2, and for each unit vector u ∈ S1, let Hu denote the discrete half plane Hu = {x ∈ Z2 : 〈x,u〉< 0}.

Definition 2. A stable direction for U is a unit vector u such that [Hu] = Hu, or in other words, one suchthat no new sites become infected when the initial set is equal to the half plane Hu. Otherwise, u is said tobe an unstable direction for U . The set S for U is the set of all stable directions for the update family U :

S := S(U ) = {u ∈ S1 | u is stable for U }.

An update rule X is said to destabilize a direction u ∈ S1 if for U = {X} one has that u 6∈ S(U ). Asmentioned earlier, it is shown in [8] that U -bootstrap percolation models can be classified into three types,depending on their update families U :• Supercritical: When there exists an open semicircle C ⊂ S1 which is disjoint from the set of stable

directions S ∩C = /0.• Critical: When S ∩C 6= /0 for every open semicircle C ⊂ S1, and there is an open semicircle C ⊂ S1

that is disjoint from Int(S ).• Subcritical: When every open semicircle in S1 has non-empty intersection with Int(S ).When studying percolation on a graph, one of the most important quantities to consider is the critical

probability pc of the model: this is the probability of initial infection at which the probability of percolationreaches 1

2 , or the threshold point. Note that percolation means that the entire graph G is infected at t = ∞,which is equivalent to saying [A0] = A∞ =V (G). Notably, through the above classification system, one canobtain insight into the value of the critical probability of infection solely based on the update family U .

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Definition 3. The critical probability of infection is defined as

pc = inf{

p | Pp(A∞ =V (G))≥ 12

}. (3)

Remark 4. In infinite graphs, Pp(A∞ = V (G)) is always 0 or 1 due to ergodicity (as percolation is atranslation invariant) [8]. In particular, this is the setting considered in [8], and thus in such cases pc =

inf{p | Pp(A∞ =V (G)) = 1} .

When considering critical and supercritical models, it is proven in [8] that pc = 0, whilst in the case ofsubcritical models, it is proven in [2] that pc > 0. In later sections, we shall consider the methods of [2] todevelop similar notation and concepts in order to address the numerous similarities between specific formsof the Trust Model in Bootstrap Percolation and subcritical U -bootstrap percolation.

1.4 Other ModelsHowever, while the activation process for the T -BP is primarily inspired by bootstrap percolation models,we had many other inspirations as well: One example is the work done in [16], which uses graphs torepresent society and considers various constraints to inhibit the spread of a rumor. In their work, theauthors only allow the rumor to spread to vertices which would be “interested” in it. In a similar fashion,our T -BP allows a rumor to spread only to vertices which are able to deem it “unbiased”. Additionally, theuse of hierarchical graphs of [16] inspired our use of those networks in Section 2.2.

From a different perspective, [9] introduces the existence of trust (and distrust) among members of asociety. However, this sense of skepticism is introduced differently in [9] than in the T -BP. Their modelbases trust on how much one trusts each person telling one the rumor, whereas the T -BP bases trust onhow many different types of people tell one the rumor (ignoring the strength of one’s relationships). Thisdifference is crucial because their model emphasizes psychological phenomena such as group-think andconfirmation bias, as one believe rumors which one’s “friends” (and not one’s foes) tell one, whereas oursis meant to combat these phenomena.

2 A Trust Model for Bootstrap PercolationReturning to the question from the introduction, we ask: How does information percolate when a messen-ger only passes the information if it has been received by a number of different sources of certain types?Motivated by this, we begin our study of the Trust Model for Bootstrap Percolation, defined in Section 1.1.

2.1 Immunity in T -BPIn the following subsections we shall focus on two forms of T -bootstrap percolation that are of particularinterest: T -BP as r-bootstrap percolation, and the simplest T -BP, which has single trust vector K. It shouldbe noted that within the T -BP, a proportion of the vertices will be immune to the disease: these verticescannot belong to At for any t ∈N unless they are in A0, as they cannot become infected from their neighbors.We shall formally define the set of immune vertices by

I :={

v ∈V (G) : ∀K j ∈T , ∃i ∈ [m] s.t. |Ni(v)|< k ji

}(4)

where T = {K1, ...,Kn} and K j = {k j1, ...,k

jm} ∈ Nm as before.

7

In order to study the T -BP on G, the vertices in I need to be removed from G, leading to a modi-fied graph which sometimes will be disconnected, with some components that might never become infecteddepending on the set A0. We shall denote by G := G−I the graph where all immune vertices and corre-sponding edges have been removed.

In order to understand the likelihood of percolation of a T -BP model and how immune vertices disruptpercolation on a network, it is useful to introduce the notion of diversity:

Definition 5. The diversity of a vertex v ∈ G in a T -BP model is the number Dv of different labels thatvertices in N(v) have:

Dv := |{i ∈ {1, . . . ,m} : Ni(v) 6= /0}| . (5)

2.1.1 Immunity for T -BP as r-bootstrap percolationIn what follows we shall study immunity of vertices for a form of T -BP known as “T -BP as r-bootstrappercolation.” The T -BP as r-bootstrap percolation model has trust vectors Ki ∈ Nm which have exactly rnon-zero entries given by 1. In this setting, the cardinality of the trust family |T | is

(mr

), and in this model,

at least r different neighbors must be infected in order for a vertex to be infected (“different neighbors”meaning neighboring vertices with distinct labels).

However, even before the infection is introduced to a T -bootstrap percolation network, there exist cer-tain vertices which will never be able to contract the infection (unless, of course, they are infected initially):these vertices are defined as immune, and we shall them describe as follows: In the case of T -BP as r-bootstrap percolation, a vertex is immune if it does not have neighbors of at least r distinct labels.

Example 6. Consider T -BP as 3-bootstrap percolation. In this case, the model can be used to represent thespread of a political rumor among a society of Democrats, Republicans, and Independents. Then, labelingthe vertices with 1,2,3 to represent each political party, suppose that, in order to limit the spread of biased(and potentially false) information, there exists a rule that an individual will only believe and pass on therumor, if he/she heard it from 2 people with different political backgrounds. Then, the trust family in thismodel would be T = {(1,1,0),(1,0,1),(0,1,1)}, and this is equivalent to 2-neighbor bootstrap percolationwith 3 labels. Yet, if an individual knows only Democrats, then it is impossible for them to get infected.Thus, they are immune.

From the definition of T -BP as r-bootstrap percolation, one can see that if a vertex is not immune,then Dv ≥ r. Hence, it is of particular interest to understand which vertices v have Dv < r, since those willcomprise all of the immune vertices.

Proposition 7. Consider T -BP on a graph G with multiple trust vectors of length m whose entries arek j ∈ {0,1}, with r non-zero. Then, for

{dj

}the Stirling number of the second kind, the probability of

immunity p0(d,m,r) for a vertex v with |N(v)|= d (equivalent to the probability that Dv ≤ r−1) is

p0(d,m,r) := P(Dv ≤ r−1) =

r−1∑j=1

[(mj

)( j!){d

j

}]md . (6)

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Proof. We shall prove (6) through the principle of inclusion and exclusion. In order to do this, we shall firstcalculate the probabilities P(Dv = i) for i = 1, . . . ,r, and use this to calculate p0(d,m,r).

Given a fixed integer n such that 1≤ n≤ r−1, in order to understand P(Dv = n) note that there are(m

n

)ways to choose the n acceptable labels that the neighbors may have, or in other words, ways to choose a setN ⊂ {1, . . . ,m} with |N| = n. Once these labels are chosen, the number of surjective functions (1.1) suchthat f (N(v)) = N is given by

n

∑i=0

(−1)n−i(

ki

)(n− i)d . (7)

To see this, we begin by noting that each of the d adjacent vertices in N(v) has n possible values for itslabel, so there are nd functions from N(v) to N. However, not all functions will be different. To visualizethis, consider a Venn diagram where each circle As of the Venn diagram corresponds to a label s, with1 ≤ s ≤ m, and is defined as As := { f : N(v)→ N : s 6∈ N}. For instance, every object in A1 will be apossible configuration of neighbors of v such that none of them are of label 1.

All functions which are not surjective to N must appear in some set Ai for i ∈ N. Therefore, the numberof onto functions f : N(V )→N is given by nd−|

⋃i∈N

Ai|. To calculate this, use the Principle of Inclusion and

Exclusion [18]. Thus, to find |⋃

i∈NAi|, consider ∑ |Ax1 ∩ ...∩Ax j | for all j such that 1 ≤ j ≤ n, and xi ∈ N,

which is given by

∑ |Ax1 ∩ ...∩Ax j |=(

nj

)(n− j)d , (8)

since there are(k

j

)ways to choose the j labels out of N that the neighbors cannot occupy, and so there are

n− j options for every neighbor’s label. Thus the number of surjective functions f : N(v)→ N isn

∑i=0

(−1)n−i+1(

ni

)(n− i)d . (9)

From the above, one can calculate the number of different label assignments that the vertices N(v) can have,where Dv = j: there are

(mj

)possibilities for j labels, which multiplied by (9) leads to(

mj

) j

∑i=0

(−1) j−i+1(

ji

)( j− i)d . (10)

Equivalently, we can write this as(m

j

)( j!){d

j

}where

{dj

}is the Stirling number of the second kind [19].

Hence, the number of different label assignments to the graph G for which the diversity Dv ≤ r−1 is

r−1

∑j=1

[(mj

)( j!){

dj

}]. (11)

Recalling that there are md possible functions f : N(v)→ {1, . . . ,m}, the probability of a vertex havingdiversity Dv ≤ r−1 is given by

P(Dv ≤ r−1) =

r−1∑j=1

[(mj

)( j!){d

j

}]md , (12)

which concludes the proof.

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Corollary 8. In the setting of Proposition 7, the number of vertices of degree d in I satisfies

|I | ≥r−1

∑j=1

[(mj

)( j!){

dj

}]. (13)

Whilst in coming sections we shall analyze the behavior of the T -BP on different graphs, the case oflattices in Zd will be of much interest, since these will illustrate some general properties of the model.2.1.2 Immunity for the T -BPThe second form of T -bootstrap percolation we shall consider is the simplest form of T -BP in which theinfection grows with a single rule K = (k1,k2, ...,km). In this setting, given any vertex v with degree d ona graph G, consider the vector x = (x1,x2, ...,xm) defined by xi = |Ni(v)|, where in particular, ∑

mi=1 xi = d.

In order for a vertex to be immune under in this setting, one must have xi < ki for some integer i such that1≤ i≤ m. We will rely on this fact for the following proposition:

Proposition 9. Consider the T -BP, with K = (k1,k2, ...,km). Then, the probability of immunity p0(d,K) fora vertex v with |N(v)|= d is

p0(d,K) = 1−d−(∑m−1

l=1 kl)

∑xm=km

(dxm

)(1m

)xm[ d−xm−(∑m−1

l=1 kl)

∑xm−1=km−1

(d− xm

xm−1

)(1m

)xm−1[· · ·

[ d−(∑ml=3 xl)−k1

∑x2=k2

(d− (∑m

l=3 xl)

x2

)(1m

)x2(

1m

)d−(∑ml=2 xl)

]...

].

Proof. Let f (a,b,c) be the function defined as

f (a,b,c) :=a−(∑i−1

l=1 bl)

∑xi=bi

(axi

)(1c

)xi[ a−xi−(∑i−1

l=1 kl)

∑xi−1=bi−1

(a− xi

xi−1

)(1c

)xi−1

· · ·

[ a−(∑il=3 xl)−b1

∑x2=b2

(a− (xi + ...+ x3)

x2

)(1c

)x2(

1c

)a−(∑il=2 xl)

]...

]where b = {b1,b2, · · · ,bi} and a,c ∈ N. Then, our claim can equivalently be stated as p0 = 1− f (d,k,m)

for k = {k1,k2, · · · ,km}. Indeed, this can be proven with an inductive argument on the number of availablelabels m.

For m = 2, the vector k = {k1,k2} satisfies k1 + k2 ≤ d. We must find the probability of assigning dvertices to one of 2 labels such that there are at least k1 vertices of label 1 and k2 of label 2–this holds whena vertex is not immune, so we must then subtract this from 1. This is equivalent to saying there may be x1

vertices of label 1 such that k1 ≤ x1 ≤ d− k2, and all other vertices of label 2. Note that the probability thatthere are x1 vertices of label 1 is

( dx1

)(12

)x1 (12

)d−x1 .

By summing over all possible values for x1 varying from k1 to d−k2, we obtain the overall p0 for m = 2

and K = {k1,k2} to be p0(d,{k1,k2}) = 1− f (d,{k1,k2},2) = 1−d−k2

∑x1=k1

( dx1

)(12

)x1 (12

)d−x1 . Now, move on

to m = 3 with k = {k1,k2,k3}. We must find the probability of assigning d vertices to one of 3 labels suchthat there are at least k1 vertices of label 1, k2 of label 2, and k3 vertices of label 3. We can approach thiswith casework:

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First, note that there may be x1 vertices of label 1 such that k1 ≤ x1 ≤ d− (k2 + k3). Then, take casesbased on the value of x1. Note that given x1, there are d− x1 remaining vertices to consider, with 2 possiblelabels to assign them to. Now this question is almost the same as the one with m= 2, with the only differencebeing that the probability a vertex occupies one of these two labels (2 or 3) is not 1

2 , but 13 . So given x1,

the probability that the number of vertices of label 2 is greater than k2 and the number of vertices of label3 is greater than k3 is just f (d− x1,{k2,k3},3). This means that given x1, the probability that the vertex isinitially not immune (we are using complementary counting) is:(

dx1

)(13

)x1

f (d− x1,{k2,k3},3). (14)

Now, as x1 can go from k1 to d− (k2 + k3), p0 for m = 3 is:

p0(d,{k1,k2,k3}) = 1−d−(k2+k3)

∑x1=k1

(dx1

)(13

)x1

f (d− x1,{k2,k3},3) (15)

= 1− f (d,{k1,k2,k3},3), (16)

which provides the intuition for the inductive step: we must prove that

p0(d,{k1, ...,km}) = 1− f (d,{k1, ...,km},m),

assuming that there exists an i such that p0(d,{k1, ...,ki}) = 1− f (d,{k1, ...,ki}, i).In order to prove the statement by induction, consider m = i+ 1, and let k = {k1,k2, · · · ,ki+1}. Using

casework once more, let the number of vertices of label 1 be x1 ∈ {k1, . . . ,d−i+1∑j=2

k j}.

Then, there are d− x1 vertices which must have labels in [i+1]/{1}. By the inductive assumption, theprobability that they have at least k j vertices of label j is f (d,{k2, ...,ki+1}, i). However, note that we mustreplace the 1

i terms in this formula with 1i+1 because the probability any individual vertex has a specific

label is now 1i+1 , meaning that it is actually f (d,{k2, ...,ki+1}, i+1). Also, the probability that there are x1

vertices of label 1 is( d

x1

)( 1

i+1)x1 , so multiplying these two terms one finds that the probability that a vertex

is initially not immune given that it has x1 neighbors of label 1 is :(dx1

)(1

i+1

)x1

f (d,{k2, ...,ki+1}, i+1). (17)

Since x1 can be anything from k1 to d−i+1∑j=2

k j, summing over the possible values of x1 and subtracting

this summation from 1 leads to the probability

p0(d,{k1,k2, · · · ,ki+1}) = 1−d−(∑i+1

l=2 kl)

∑x1=k1

(dx1

)(1

i+1

)x1

f (d,{k2, ...,ki+1}, i+1) (18)

= 1− f (d,{k1, ...,ki+1}, i+1). (19)

This concludes the inductive step, finalizing the proof.

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2.1.3 A Network’s ImmunityThe formulas for p0 from the previous sections relate only to single vertices. In order to apply this conceptto entire graphs, we look at the expected fraction of immune vertices on any given graph, as a function of itsdegree distribution P(d), which is defined to be the fraction of nodes in the graph with degree d.

Definition 10. We shall denote by p0(G,T ) the expected fraction of immune vertices for a T -BP on agraph G, and by p0(d,T ) the probability of immunity for a vertex of degree d in G.

Corollary 11. In a T -BP on a graph G with degree distribution P(d), one has that the expected fraction ofimmune vertices is

p0(G,T ) =∞

∑j=0

p0( j,T )P( j). (20)

2.2 Immunity on Hierarchical Networks

Figure 2: The network being built at each step n.

After finding the general formula for p0(G,T ) inthe previous section, we focus our study to hierar-chical networks. We wish for the graphs we applythe T -BP on to represent society. A notable qualityof many social networks is that they have a power law degree distribution, which means that the fractionP(d) of vertices in G having degree d is approximately P(d) ∼ d−γ for some constant γ ∈ R. For socialnetworks particularly, γ is often around between 2 and 3. However, while social networks are relativelyrandom, graphs known as deterministic hierarchical networks have power law degree distributions whilealso having predetermined configurations, making them “simpler” to study. Thus, it would be interesting toinvestigate the T -BP on a deterministic hierarchical network with γ being approximately 2 to 3.

One such example is the graph in Figure 2, defined in [6]: Start with one node, a root node. Addtwo more nodes and connect them to the root. At step n, add 3n−1 nodes each, identical to the figure inthe previous iteration (step n− 1) and connect the 2n bottom nodes to the root. The degree exponent isγ = 1+ ln3

ln2 . From this, and using results from [6], we discover the following proofs:

Lemma 12. At n iterations, there are 2(3n−log2(d+2)) vertices of degree d.

Proof. From [6], at n iterations, there are(2

3

)3n−i vertices with degree 2i+1− 2. Then, by substituting d

with 2i+1−2 and solving, we find that there are 2(3n−log2(d+2)) vertices of degree d at this step.

Corollary 13. The probability a vertex has degree d is 2(d +2)− log2 3.

Proof. There are also 3n vertices total at n iterations. So the probability a vertex has degree d at n iterationsis 2(3n−log2(d+2))

3n , or 2(3− log2(d+2)), which can be simplified to 2(d +2)− log2 3.

Proposition 14. The expected fraction of immune vertices p0(G,T ) for this hierarchical network is2n+1−2

∑d=1

2(d +2)− log2 3 p0( j,T ).

12

Proof. According to the paper, the maximum degree of any vertex on this graph is the degree of the root,which at step n is 2n+1− 2. Hence, this is the upper bound of the summation. To find p0, or the expectedfraction of initially immune vertices on this model with update vector set L, we need to find the probabilityof immunity for a vertex with j neighbors and then multiply that by the probability that a vertex has jneighbors. Finally, we need to sum this product across all possible j. Using this method, we find that

p0(G,T ) =2n+1−2

∑d=1

2(d +2)− log2 3 p0( j,T ).

2.3 Critical ProbabilityHaving studied the immune vertices of the graph G in the previous section, we shall consider now a T -BPfor which the set A of initially infected vertices is chosen randomly from G, by setting the probability ofbeing initially infected to be p. To understand this model, consider the critical probability pc of infectionfor which the probability of percolation is 1

2 .

Proposition 15. In a T -BP on a graph G, the critical probability of infection satisfies pc ≥ 2p0−12p0

.

Proof. The probability that there exists an immune vertex that is not initially infected is p0(1− pc). Ifp0(1− pc) >

12 , then the probability of percolation is already less than 1

2 , as the existence of an uninfecteddead vertex prevents percolation. So, a lower bound for the critical probability pc is 2p0−1

2p0.

3 Two-Dimensional Trust Model for Bootstrap PercolationFollowing the more general results, the primary focus of our paper lies on the T -BP for m= 2 as 2-bootstrappercolation on Z2. This model has K = ((1,1)). The primary motivations for using the graph Z2 are becausenew bootstrap percolation models are often first studied on the 2D lattice and also because many of ourresults are inspired by the past study of U -bootstrap percolation, which has only been studied on Z2.

3.1 Critical probabilityRecall the definition of pc, the critical probability of infection. In [8], the authors prove that pc = 0 for criti-cal and super-critical models. The basis of their argument is that the lack of stable directions in these modelsallows for almost unrestrained growth of the infection from multiple directions, ultimately enabling percola-tion. In contrast, the argument made in [2] proves that pc > 0 when the update family, U , is subcritical. Asa very general outline of the proof, the idea is that since so many directions are strongly stable-in particular,there are 3 such that the triangle formed by these 3 directions contains the origin-all growth will be boundedby triangles with side lengths perpendicular to these strongly stable directions if p is small enough, thuspreventing percolation.

Our ultimate goal would be to prove the existence of similar “triangles” on T -BP for m = 2 as 2-bootstrap percolation, and we conjecture that, similarly to subcritical U -bootstrap percolation,

Conjecture 16. For 2-bootstrap percolation with m = 2 on Z2, pc > 0.

Thus, in the following sections, although we will study various specific models of 2-bootstrap perco-lation with m = 2 and derive results for each of these models, the motivation will be to develop a greaterintuition for the general 2-bootstrap percolation model, along with terminology which will hopefully beuseful in approaching our conjecture.

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3.2 T -BP as U -Bootstrap PercolationWe shall dedicate this section to the study of a network on which T -BP appears as a form of U -bootstrappercolation.

3.2.1 Striped T -BP

Figure 3: The lat-tice configuration forstriped BP, with ver-tices of label 1 (black)and 2 (white).

Begin with Z2 and assign to the vertices such that the graph appears “striped”. Morerigorously, assign all vertices with even x-coordinate to label 0, and the remainingvertices to label 1. Now, the rules of T -BP for m = 2 as 2-bootstrap percolationstill apply: if a vertex has 2 of its neighbors with different labels infected, then it toobecomes infected. So, note that for any vertex on this graph, its vertical neighbors(the neighbors above and beneath it on the lattice) have the same label, while itshorizontal neighbors have the other label. This gives a fixed set of update rules forall the vertices, defining U as follows:

Figure 4: The update family U for striped BP. There are 4 rules, shown here.

Lemma 17. The update family U for striped bootstrap percolation is critical.

Proof. The first step is to find the stable directions. We claim that they are {±1,0} and {0,±1}. It is easyto verify that these work. Additionally, these four stable directions are enough to fulfill the first criteria: thatthere must exist a stable direction in every open semicircle C ⊂ S1. Now, we must show that all other u ∈ S1

are not stable. Let us first focus on vectors in the first quadrant. The proof can be extended to the otherquadrants by symmetry. So, in the first quadrant, the border of any Hu will have the following components:

Figure 5: The configurations seen on the border of Hu for unit vectors u in the first quadrant.

Having the first, diagonal, component defined in Figure 5 at some point on the boundary will result ina newly infected site from the fourth rule in U . Therefore, all u in the first quadrant are unstable. And bythe symmetry of the model, it is clear that the same holds for every quadrant–all u not on the axes will havethe “diagonal components”, and therefore will be unstable. Thus, the only stable directions are the axes.Additionally, this model lacks any strongly stable directions, making it critical.

3.2.2 Diagonal BPThe next model is another form of 2-bootstrap percolation with m = 2, which again happens to be a form ofU -bootstrap percolation. This we call diagonal bootstrap percolation.

14

Figure 6: The latticeconfiguration for diag-onal BP. Black verticesare label 1, white are la-bel 2.

Begin with Z2 and assign labels 1 and 2 to the vertices such that the graphappears “diagonally striped”. Essentially, consider lines of the form y = x+ b, forb ∈ Z. If b≡ 0,1 mod 4, make all vertices on the line have label 1, and otherwise,assign them label 2. Next, note that for any vertex on this graph, its neighbors upand to the left have the same label, while its neighbors down and to the right havethe other label. This gives a fixed set of update rules for every vertex, which definesU for this model:

Figure 7: The update family U for diagonal BP. There are 4 rules, shown here.

Lemma 18. The update family U for diagonal bootstrap percolation is subcritical.

Proof. A subcritical update family is one such that for every open semicircle C ⊂ S1, there is a stronglystable direction u such that u ∈C. Now, we claim that every u in quadrants 2 and 4 are strongly stable underthis update family U –which would then satisfy the claim that it is subcritical.

First, it is clear that the axes are stable directions. Next, look at all unit vectors u in the second quadrant.Note that any such u will create an Hu with the components similar to those Figure 5 somewhere on theborder. The only difference is that the first component (the diagonal one), is reflected across the y-axis.Clearly, then, no u in the second quadrant will result in new vertices being infected, because X3 and X4 bothwill not destabilize any u in the second quadrant. A similar argument can be applied to the fourth quadrant,and ultimately we find that all u in quadrants 2 and 4 are stable, making U subcritical.

3.3 Other two dimensional T -BPIn what follows we shall introduce two different networks on which the T -BP does not appear as a form ofU -bootstrap percolation.

3.3.1 3-1 BP

Figure 8: The latticeconfiguration for 3-1BP. Black vertices arelabel 1, white are label2.

Next, we investigate a different form of 2-bootstrap percolation with m = 2 which isnot a form of U -bootstrap percolation. This we call 3-1 percolation, and because itis our first example of a model that is distinctly part of T -BP, but not U -bootstrappercolation, we use it to define terminology useful to describing T -BP overall. Be-gin with Z2 and assign labels to the vertices such that the graph appears as shown inFigure 8.

Lemma 19. 3-1 bootstrap percolation is not a form of U -bootstrap percolation.

Proof. U -bootstrap percolation is defined to be a form of bootstrap percolation where every vertex in thegraph G obeys the same set of update rules, U . However, note that for 3-1 BP, not all vertices have the sameupdate rule. There are two possible U here:

15

Figure 9: The update families U1 and U2 for 3-1 BP. There are 3 rules for each U , shown here.

Thus, 3-1 bootstrap percolation is not a specific case of U -bootstrap percolation.

Because T -BP can have multiple update families U , we must create a version of stable directions thatapplies to all T -BP:

Definition 20. The direction u is a d-stable direction if there exists an i ∈ [d] such that [Hu + i] = Hu + i ,where Hu + i indicates a fixed half-plane Hu shifted to the right i units. A direction u ∈ S1 is a stronglyd-stable direction if u ∈ I, where I ⊂ S1, such that for all directions s ∈ I, s is d-stable.

Definition 21. A graph is d-critical if the set of directions D that are k-stable (where k ≤ d), satisfy that forevery open semicircle C⊂ S1, C∪D 6= /0 and there exists a semicircle C′ ⊂ S1 such that there are no stronglyd-stable directions in C′. A graph is d-subcritical if every open semicircle C ⊂ S1 contains a stronglyd-stable direction.

Lemma 22. The 0-stable directions for 3-1 bootstrap percolation are the axes. Moreover, every otherdirection u ∈ S1\{i, j,−i,− j} is 1-stable. Thus, 3-1 BP is 1-subcritical and 0-critical.

Proof. The first part is clear-simply check every element of {i, j,−i,− j} for 0-stability. Next, we focus onvectors in the first quadrant only. The proof can be extended to the other quadrants by symmetry. So, inthe first quadrant, the border of any Hu will have the components from Figure 5. The diagonal componentsimply that first, if the vertices that form the diagonal components are of the same label, then no new siteswill be infected. And if the diagonal vertices are of different labels, new sites will be infected (becauseof the second update rule in U1), but translate the boundary by 1 and now they have the same label–so[Hu +1] =Hu +1. Thus, the directions not on the axes all 1-stable.

3.3.2 Checkerboard BP

Figure 10: Lattice con-figuration for checker-board BP. Black ver-tices are label 1, whiteare label 2.

Begin with Z2 and assign labels to the vertices such that the graph appears as shownin Figure 10. One may note that there are two possible update families which areboth subcritical: S1 is quadrants 2 and 4, while S2 is quadrants 1 and 3:

Figure 11: The update families U1 and U2 for checkerboard BP, with 4 update rules for each.

Lemma 23. The 0-stable directions for checkerboard bootstrap percolation are the axes. Every other di-rection u ∈ S1\{i, j,−i,− j} is 1-stable. Thus, checkerboard BP is 1-subcritical and 0-critical.

16

Proof. This proof is nearly identical to the one for Lemma 22. The first part is clear–simply check everyelement of {i, j,−i,− j} for 0-stability. Next, we focus on vectors in the first quadrant only. The proofcan be extended to the other quadrants by symmetry. So, in the first quadrant, the border of any Hu willhave the components from Figure 5. The diagonal components imply that first, if the vertices that form thediagonal components are of the same label, then no new sites will be infected. And if the diagonal verticesare of different labels, new sites will be infected (because of the second update rule in U1), but translate theboundary by 1 and now they have the same label–so [Hu +1] =Hu +1. Thus, the directions not on the axesall 1-stable.

3.3.3 Mixed BPWe study this model in order to gain insight into less regular models. The network coloring is shown inFigure 13 (Right). The possible update families are:

Figure 12: The update families U1, U2, U3, and U4 for mixed BP. There are 4, 4, 3, and 3 rules, respectively, for each.

Note the stable directions for each U : the family U1 has stable directions in quadrants 2 and 4; thefamily U2 has stable directions in quadrants 1 and 3; the family U3 has stable directions in quadrants 3 and4; and finally the family U4 has stable directions in quadrants 1 and 2. In Figure 13 (Left) one can see eachvertex by which update family Ui it has.

Figure 13: Left: Red vertices are U1, yellow are U2, green are U3, and purple are U4. Right: the coloring of the mixedBP, where black vertices are label 1, white are label 2.

To understand the stable directions, consider some unit vector u in the first quadrant. The vertices thatcan be newly infected from Hu must have update families U1 or U3, as quadrant 1 is unstable for thesefamilies. By symmetry, it can be seen that as long as a unit vector u has Hu with a boundary such that thereare 3 consecutive diagonal vertices, it is 2-stable. An obvious example would be u = (

√2

2 ,√

22 ). An example

is displayed in Figure 14, where the process of infection is shown:

17

Figure 14: The left lattice contains the vertices labeled by their update family and the right has vertices labeled bytheir state (as in Figure 13). Suppose the vertices surrounded by gray-filled circles are initially infected. The newlyinfected vertices are surrounded by black circles, and note that they have update families U1 and U3, as both of thesefamilies are unstable for u in the first quadrant.

3.4 Trust Model for T -BP for m = 2 as 2-bootstrap percolation on Z2

So now we turn to the more general model: T -BP for m = 2 as 2-bootstrap percolation on Z2, with randomassigning of labels to vertices. First, we want to be able to predict whether its growth is relatively unbounded,as in critical and supercritical models, or restricted, as in subcritical models.

Let us consider all possible update families U on the model. The reason that this model does notalready fall under the umbrella of U -bootstrap percolation is because each of its vertices obeys a differentset of update rules depending on the configuration of its neighbors, making our model non-homogeneous.Therefore, we can ask whether each possible U , based on an individual vertex, is critical or subcritical. Wediscover that out of all possible U on this model, only 1

8 are critical-the rest are subcritical. From here,we arrive at the main Conjecture 16 from Section 3.1. We sketch here an approach that may allow us toultimately prove Conjecture 16.

Definition 24. A finitely stable graph is one such that there exists an integer s such that every directionu ∈ S1 is s-stable.

Conjecture 25. Every finitely stable graph has pc > 0.

We hope to approach this problem with a method similar to that used in [2], as the growth of the infectionwould hopefully be bounded by shapes with sides perpendicular to the finitely stable directions of the model.We also define:

Definition 26. A critical-free network is a network in which the update family for every individual vertexis subcritical.

Conjecture 27. Every critical-free network has pc > 0.

Then, if a critical-free network has pc > 0, this might be used to show that T -BP with m = 2 as 2-bootstrap percolation on Z2 in general has pc > 0. This is because since the probability of a vertex beingcritical is only 1

8 , we believe the general model is sufficiently similar to the critical-free network that it wouldhave pc > 0 for similar reasoning.

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4 T -BP on Random NetworksThus far, we have studied T -BP on fairly regular graphs, mainly for simplicity. However, society is rarelythis regular-which is why for this section, we focus on Erdos-Renyi graphs, or random graphs. The resultsare primarily analytical, and the codes can be found in the appendix. In particular, we study the followingproperties if the model:• The initial probability of infection p;• The critical probability of infection pc;• The “fraction percolated” (the fraction of graph infected by A∞)• The probability of having an edge between two vertices, denoted den, short for density.

These probabilities are considered in terms of the main variables of the T -BP model: the time t; the numberm of labels a vertex may have; and the number r of different labels the set of infected neighbors must includein order for a vertex to be infected. Additionally, throughout this section, for any value requiring multipletrials, we run 30 trials. Additionally, we always look at graphs of the same size, having n (the number ofvertices, |V (G)|) equal to 10000.

4.1 Varying Density

Figure 15: This shows how the fraction percolatedvaries as p varies given various den. Note that n =

10000, m = 3, r = 2. The different densities consid-ered are shown by the differently colored curves.

The first relationship we consider is the one between thefraction of a graph that would end up percolated and theinitial probability of infection, p. We analyzed this fora graph with m = 3 and r = 2, with various values forden: 0.0005,0.001,0.002,0.003,0.004,0.005. The graphis shown in Figure 15. Notably, the metastability effect(where a slight shift in p causes a large jump in the frac-tion percolated) is evident in this model.

Looking at an individual curve in Figure 15, one candetermine the critical probability pc for a network with m = 3,r = 2, and some fixed value for den. There-fore, we look at a more consolidated version of the results, as we compare the value of pc to the value ofden. We also expand our investigation to different values of m and r: we set (m,r) to be (3,2),(4,2),(4,3),and find the graph for pc compared with den in each case, as shown in Figure 16.

Figure 16: This figure shows how pc varies as den varies. Note that n = 10000 for all cases. The pair (m,r) from leftto right is (3,2),(4,2),(4,3).

Interestingly, this seems to follow a power law curve rather precisely. For each pair (m,r), the curve thatbest fits is

19

• (m,r) = (3,2): pc = 0.000000019568213(den)−1.897075330420950, with correlation coefficient r2 =

0.996.• (m,r) = (4,2): pc = 0.000000027712517(den)−1.833916660925080, with r2 = 0.995.• (m,r) = (3,2): pc = 0.0000004688(den)−1.7465029493, with r2 = 0.993.

4.2 Varying the number of required labels

Figure 17: This shows how the fraction per-colated varies as p does, for various r. Note,n = 10000, m = 4, den = 0.005.

We once again look at the relationship between the fraction ofa graph that would end up percolated and the initial probabil-ity of infection, p, but while varying the number of requiredlabels, r. We begin with a graph with m = 4 and den = 0.005,with various values for r: 2,3,4, shown in Figure 17.

Notably, the metastability effect is clear here as well. Asin the previous section, we look at each individual curveand determine the value of pc for each r. Next, we plotr against pc, for various pairs of parameters (m,den) set to(4,0.005),(4,0.003),(5,0.003) in Figure 18.

Figure 18: This figure shows how pc varies as r varies. Note that n = 10000 for all cases. The pair (m,den) from leftto right is (4,0.005),(4,0.003),(5,0.003).

Here, the relationship is more difficult to determine, as m and r are both small integers, and so there areonly a small number of data points available to calculate. Nonetheless, it appears to fit a power law curveagain, albeit more weakly in some cases. For each pair (m,den), the curve that best fits is• (m,den) = (4,0.005): pc = 0.0000135564r5.3410975882, with r2 = 0.997.• (m,den) = (4,0.003): pc = 0.0000030675r8.0395325660, with r2 = 0.975.• (m,den) = (5,0.005): pc = 0.0000390111r4.8654339012, with r2 = 0.994.Overall, pc seems to have a power law relationship with both den and r. This suggests that perhaps the

equation for pc is of the form pc = c f (m)re1den−e2 , where c,e1,e2 are all positive constants. Also, varyingm affects pc directly, which is why the f (m) component is present.

5 ConclusionBootstrap percolation has been used for years to model various percolation processes, with applicationsspanning from epidemiology to rumor spreading. The Trust Model in Bootstrap Percolation has been de-veloped to add an extra layer to the current forms of bootstrap percolation, as the vertices of the graphson which the infection spreads are now labeled. The T -BP was originally created to represent the spreadof information based on the basic human instinct to trust information more if it comes from a variety ofsources.

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Using the T -BP, we first looked at the probability of a vertex being immune. We discovered p0, or theprobability of immunity, for vertices on various models: the T -BP as r-bootstrap percolation, the simplestform of T -BP with only one trust vector, and T -BP on the deterministic hierarchical graph defined insection 2.2. Following this, we discovered the expected number of vertices which will be immune on anyparticular graph with these models. We also offer a lower bound for pc, the critical probability of infection,based on p0. In Section 3, we looked to the T -BP with m = 2 as 2-bootstrap percolation model on Z2.By investigating specific “colorings” of this model (equivalently, by assigning labels to the vertices onZ2 in a non-random fashion), we were able to prove their criticality and define terminology and conceptswhich will hopefully be of use in proving the conjecture that pc > 0 for this model. Finally, we concludedour investigation by looking to random graphs (as these better represent the irregularities of society) andderiving analytical results on these graphs by running the T -BP model on them computationally.

It would be of interest to explore the T -BP on different graphs. Particularly, deterministic hierarchicalgraphs seem promising for applications in sociology and marketing, as society has the same power lawdegree distribution as these graphs do. Progressing further, one may study the T-BP on random graphs withthis power law degree distribution, and eventually make the model stochastic by introducing probability ofthe infection spreading from one vertex to another.

Additionally, the T-BP has several applications. A prevalent issue today is that of echo chambers. Socialmedia networks and search engines keep track of news a user responds positively to, and use this informationto suggest future articles and advertisements. However, this means one will increasingly only see materialthat is in line with one’s stated interests. This worsens issues of polarization and group-think. To combatthis phenomenon, one might apply the T -BP to these news-presenting algorithms. In turn, it would beinteresting to acquire data from actual social networks and apply the model to it, as was done in [15]. Thus,we might ask questions such as: does the T -BP prevent the spread of fake or highly biased news? For thenews that does manage to percolate through society, would it slow it down, or speed it up?

Finally, another interesting application to this model is in the study of the spread of genetic diseases.Apply the T -BP to directed binary trees–representative of family trees and use 2 labels, where each labelrepresents a possible sex. In order for a vertex to be infected, it must be infected by two vertices above it,where one vertex is a male and one is female. This represents how dominant X-linked genetic diseases areonly passed on if both the male and female parent have the disease. For recessive X-linked diseases, themodel would need to have 3 labels- an infected female, a female carrier, and an infected male, and wouldneed to be stochastic (as the probability of infection from a female carrier and infected male would only be12 ).

6 AcknowledgementsThis project was completed under MIT-PRIMES-USA, which the authors would like to thank for this oppor-tunity. We would also like to thank Dr. Tanya Khovanova for her invaluable advice and Svetlana Makarovafor her helpful feedback.

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